Omar Zoghlami
Geometry of the sub-Lorentzian Heisenberg group
The sub-Lorentzian Heisenberg group is a non-holonomic analogue of a spacetime: motion is confined to a two‑dimensional horizontal distribution endowed with a Lorentzian metric, which induces causal and timelike directions and furnishes a minimal model for constrained relativistic dynamics. Adopting a variational viewpoint, we show that length‑maximizing geodesics are precisely the horizontal lifts of solutions to a Lorentzian isoperimetric problem in the Minkowski plane—straight timelike segments, broken null lines, or timelike hyperbolae, depending on the enclosed signed area. We further compute that the Lorentzian Hausdorff dimension of this space is 4 and that the corresponding measure agrees, up to a positive constant, with the Haar (3‑dimensional Lebesgue) measure. In the synthetic optimal‑transport framework, we show that the space admits no timelike curvature lower bounds: contrary to the positive signature case, neither $\mathsf{TCD}(K,N)$ nor $\mathsf{TMCP}(K,N)$ hold for any $K\in\mathbb{R}$ and $N\ge 1$. Finally, we classify boost‑symmetric constant‑mean‑curvature surfaces up to ambient isometry, identifying them as lifts of Minkowskian constant‑curvature curves and as a natural class of solutions to Lorentzian isoperimetric problems.
Jona Röhrig
Lorentzian Ollivier Ricci Curvature
We adapt the notion of Ollivier–Ricci curvature to the Lorentzian setting, obtaining a synthetic notion of Ricci curvature for Lorentzian metric measure spaces and causal sets.
Tobias Beran
Coordinates for Lorentzian pre-length spaces with curvature bounded below
Abstract TBA
Marcos Ricardo Cavicchioli de Almeida
From actions to isometries: an equivalence problem for semidirect product Lie groups
Determining whether two semi-Riemannian manifolds are isometric is perhaps the most natural problem in geometry. In a Lie group endowed with a left-invariant metric, one can work at the Lie algebra level by considering a nondegenerate symmetric bilinear form. Thus, the equivalence problem for metric Lie groups can be studied in terms of metric Lie algebras. Here, we consider a particular class of metric Lie algebras: a semidirect product of the form $\mathbb{R}^n \ltimes \mathfrak{h}$, where the action of $\mathbb{R}^n$ satisfies certain properties with respect to the metric. The special case in which $\mathfrak{h}$ is nilpotent and the decomposition is orthogonal has been extensively studied in the Riemannian setting, since these Lie algebras play a central role on Einstein solvmanifolds. In those works, it was investigated whether the action of $\mathbb{R}^n$ could be assumed to be self-adjoint, and, under certain hypotheses, the answer was positive. With this result, it became possible to establish a correspondence between Einstein Riemannian solvmanifolds and algebraic Ricci solitons on nilpotent Lie groups. This question has recently been addressed in the indefinite case, under almost the same hypotheses as in the positive-definite setting. In this work, we focus on the equivalence problem in the semi-Riemmanian case. We drop the assumption that $\mathfrak{h}$ is nilpotent and no longer require the decomposition to be orthogonal, and we investigate what can occur when considering the self-adjoint part of the action of $\mathbb{R}^n$. Our goal is to determine the necessary hypotheses under which this process defines a new Lie algebra and, in such cases, when this metric Lie algebra is isometric to the original one. We provide examples showing that the answer can be positive, as well as others illustrating the difficulties involved, since a Lie algebra endowed with an indefinite metric does not need to be geodesically complete. This presentation is part of an ongoing Master's project, supervised by Viviana del Barco and a current research stay abroad under the supervision of Ana Cristina Ferreira.
Alfonso García-Parrado
Gravitational radiation and the evolution of gravitational collapse in cylindrical symmetry
Using the Sparling form and a geometric construction adapted to spacetimes with a 2-dimensional isometry group, we analyse a quasi-local measure of gravitational energy. We then study the gravitational radiation through spacetime junctions in cylindrically symmetric models of gravitational collapse to singularities. The models result from the matching of collapsing dust fluids interiors with gravitational wave exteriors, given by the Einstein–Rosen type solutions. For a given choice of a frame adapted to the symmetry of the matching hypersurface, we are able to compute the total gravitational energy radiated during the collapse and state whether the gravitational radiation is incoming or outgoing, in each case. This also enables us to distinguish whether a gravitational collapse is being enhanced by the gravitational radiation.
Javier Hervás Aniorte
Spacelike Brakke flows
We develop a weak theory of spacelike surfaces moving by its mean curvature.
Thijs de Kok
Pontryagin class obstructions to the existence of metrics with purely electric or purely magnetic Weyl curvature tensors
Do all manifolds that admit Lorentzian metrics also admit such metrics that have a purely electric or purely magnetic Weyl curvature tensor? To (partially) answer this question, we derive the vanishing of all products of Pontryagin classes that land in the top-degree de Rham cohomology of a 4k-dimensional pseudo-Riemannian manifold with a PE or PM Riemann or Weyl curvature tensor. For compact manifolds, this gives nontrivial cohomological obstructions to the existence of such pseudo-Riemannian metrics. These obstructions can be linked to the existence of Lorentzian metrics of several Petrov subtypes, which play an important role in classifying exact solutions to the Einstein equations. Moreover, they can be applied to foliations by nondegenerate umbilic hypersurfaces, which may appear as timeslices of spacetimes.
Benachour
Weighted flat translation surfaces in minkowski 3-space with density
In this work, we investigate flat translation surfaces in the 3-dimensional Minkowski space endowed with a radial density. We provide a classification of such surfaces, emphasizing the interplay between the geometry induced by the Minkowski metric and the weighted measure determined by the density. Our results extend previous studies on translation surfaces in classical spaces to the weighted Lorentzian setting.
Mikale Reddy
The BKL Conjecture and T2 Symmetric Spacetimes
The BKL conjecture (Belinskii-Khalatnikov-Lifshitz) proposes that the ap- proach to a spacelike singularity in general relativity is governed by local, oscil- latory and independently evolving dynamics at each spatial point — a picture often described as “mixmaster” behavior. Despite decades of heuristic and numerical support, a rigorous mathematical treatment remains largely out of reach, even in symmetry-reduced settings. T2-symmetric spacetimes — vacuum spacetimes admitting a two-dimensional abelian isometry group acting on spacelike orbits — provide a natural arena in which to begin making these ideas precise. In this talk, we introduce the BKL conjecture and describe the hier-archy of symmetric spacetime models through which one hopes to approach it. We discuss what is known in the simpler polarized and Gowdy subclasses, the additional difficulties that arise in the full T2-symmetric setting, and how one might attempt to extract rigorous asymptotic information in this context. The talk is intended to motivate an ongoing research program rather than report on completed results.
Irving Hernández Rosas
The Witten proof of the positive mass theorem
In this poster we explain with detail the Riemannian case of the proof of Edward Witten using spin geometry, and giving a glance of how to use it to prove the Lorentzian case. Finally we show how this techniques are used today in mathematical relativity and in what kind of problems.